User blog:Granpa/Brown dwarfs and degenerate matter
﻿The pressure that an electron orbital can resist without becoming degenerate = 1/radius^6. (Except for nondegenerate matter which is 64 times greater) Therefore the maximum radius that a solid astronomical object can become before its core collapses depends only on its atomic mass. This is why all planets from 1 to 52 jupiter masses are approximately the same size Atoms of degenerate matter are four times smaller than non degenerate matter. Double degenerate atoms are eight times smaller. Triple degenerate are 16 times smaller. Image shows 1 diatomic hydrogen molecule next to 2 monatomic helium atoms A hydrogen molecule and a helium atom both have a charge of 2 and are therefore the same size. Outer electron shell is triple degenerate. Inner electron shell is quadruple degenerate Hydrogen Fusion begins at triple degenerate (52 Jupiter masses) Helium Fusion begins at quadruple degenerate (416 Jupiter masses) You should be able to see why diatomic hydrogen fuses at triple degenerate while monatomic helium does not If the number of protons and neutrons are the same then 2 quadruple degenerate atoms can fuse together and produce one triple﻿ degenerate atom of the same size.﻿ At any given temperature hydrogen and deuterium have the same kinetic energy but deuterium has sqrt(2) times as much momentum. n)The layers within the 52 JM (jupiter mass) protosun, their thickness in km, their mass, ::their densities in g/cm^3, ::::the pressures at which each layer is expected to collapse, and the approximate mass at which that pressure is achieved are as follows: When the core finally collapsed its outer layers were spun off into orbit and became the outer planets 1)Nondegenerate hydrogen and helium, 5900km, 7.015 EM (became planet X?) ::0.071 g/cm^3 and 0.125 g/cm^3 (average=0.0794) ::::5 million bar 2)Degenerate metallic hydrogen and helium, 5900km, 388 EM (became Neptune/Uranus/Saturn/Jupiter) (96.6 g's) ::4.544 g/cm^3 and 8 g/cm^3 (average= 5.0816 g/cm^3) (6 g/cm^3 if solid) ::::320 million bar at 7 Jupiter mass 3)Double degenerate metallic hydrogen and helium, 78,460km, 52 JM (16,527 earth masses) (109.1 g's) ::36.35 g/cm^3 and 64 g/cm^3 (average= 40.65 g/cm^3) (48.78 g/cm^3 if solid) ::::20.48 billion bar at 52 Jupiter masses 4)Triple degenerate metallic hydrogen and helium, 0km (becomes 12,000 km after collapse) ::291 g/cm^3 and 512 g/cm^3 (average= 325 g/cm^3) (384 g/cm^3 if solid) ::::1.31072 trillion bar at 0.5 solar masses Radius of Jupiters core = 44,000 km (69900 km) (6.3 g's) ::5.0816 g/cm^3 * 4*pi*(44000 km)^3/3 in earth masses = 303.6 (+17 earth masses of hydrogen) Radius of Saturns core = 29,000 km (57300 km) ::5.0816 g/cm^3 * 4*pi*(29000 km)^3/3 in earth masses = 87 (+10 earth masses of hydrogen) :::: Equations used: n=degeneracy level (-1 for nondegenerate matter. 1 for degenerate. 2 for double degenerate) Radius of ground state orbital = 1.8 angstroms/(atomic number * 2^n) Density = x*0.03125 g/cm^3 * (atomic mass)/radius^3 = 0.03125 g/cm^3 * atomic mass*(atomic number * 2^n)^3 ::(Use 2 for atomic mass and atomic number of hydrogen because its diatomic.) Pressure required for collapse = 1/radius^6 = 36,621 bar * (atomic number * 2^n)^6 ::This explains why all brown dwarfs are approximately the same size ::Radius = longest dimension of orbital ::x=0 for nondegenerate matter Maximum size of planet = 150,000 km/atomic mass Mass = 1.333 * pi * size^3 * density ::1.333 * pi * (150000 km/atomic mass)^3 * x * 0.03125 g/cm^3 * atomic mass * (atomic number * 2^n)^3 Earth reaches 1.5 million bar at 3000 km ::3000km * (5g/cm^3) * (9.8m/s^2) in bar = 1.5 million bar﻿ Examples: Pressure in center of jupiter sized sphere of hydrogen ::G * (0.0794 g/cm^3)^2 * (4/3) * 3.14159 * (76000 km)^2 / 2﻿ in bar﻿ = 50,900 bar Pressure in center of jupiter sized sphere of metallic hydrogen ::G * (6 g/cm^3)^2 * (4/3) * 3.14159 * (80000 km)^2 / 2﻿ in bar﻿ = 322 million bar Pressure in center of jupiter sized sphere of double degenerate metallic hydrogen ::G * (48 g/cm^3)^2 * (4/3) * 3.14159 * (80000 km)^2 / 2﻿ in bar﻿ = 20.6 billion bar Pressure in center of jupiter sized sphere of triple degenerate metallic hydrogen ::G * (384 g/cm^3)^2 * (4/3) * 3.14159 * (80000 km)^2 / 2﻿ in bar﻿ = 1.32 trillion bar Pressure in center of half jupiter sized sphere of metallic deuterium (atomic mass=2) ::G * (2*6 g/cm^3)^2 * (4/3) * 3.14159 * (40000 km)^2 / 2﻿ in bar﻿ = 322 million bar﻿ Category:Blog posts